@Article{Rey2020,
  Title   = {Another counterexample to {Z}ygmund's connjecture},
  Author  = {Guillermo Rey},
  Journal = {Proc. Amer. Math. Soc.},
  Year    = {2020},
  Pages   = {5269--5275},
  Volume  = "148"
}
        

@article{BCOR2019,
title = "Sparse domination and the strong maximal function",
journal = "Advances in Mathematics",
volume = "345",
pages = "1 - 26",
year = "2019",
issn = "0001-8708",
doi = "https://doi.org/10.1016/j.aim.2019.01.007",
url = "http://www.sciencedirect.com/science/article/pii/S0001870819300325",
author = "Alex Barron and José M. Conde-Alonso and Yumeng Ou and Guillermo Rey",
keywords = "Biparameter, Multiparameter, Sparse domination, Maximal functions",
abstract = "We study the problem of dominating the dyadic strong maximal function by (1,1)-type sparse forms based on rectangles with sides parallel to the axes, and show that such domination is impossible. Our proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance."
}
        

@article {DSLR,
author    = {Domingo-Salazar, Carlos and Lacey, Michael and Rey, Guillermo},
title     = {Borderline weak-type estimates for singular integrals and square functions},
journal   = {Bulletin of the London Mathematical Society},
volume    = {48},
number    = {1},
publisher = {Oxford University Press},
issn      = {1469-2120},
url       = {http://dx.doi.org/10.1112/blms/bdv090},
doi       = {10.1112/blms/bdv090},
pages     = {63--73},
year      = {2016},
}
        

@Article{Rey2016,
  Title   = {On the embedding of ${A}_1$ into ${A}_\infty$},
  Author  = {Guillermo Rey},
  Journal = {Proc. Amer. Math. Soc.},
  Year    = {2016},
  Pages   = {4455--4470},
  number  = "10"
}
        

It was shown by Andrei Lerner that Calderón-Zygmund operators could be pointwise-bounded by a sum of 'positive dyadic shifts', weighted by an exponentially decaying factor: \[ |Tf(x)| \lesssim \sum_{m=0}^\infty 2^{-m \delta} \mathcal{A}_{\mathcal{S}}^m |f|(x), \] for almost every $x \in Q$, where $Q$ is some arbitrary cube in $\mathbb{R}^d$. Here the factor $\delta$ is a number in $(0,1)$ which depends on the regularity of the operator $T$.

The operator $ \mathcal{A}_{\mathcal{S}}^m$ is defined as follows: \[ \mathcal{A}_{\mathcal{S}}^m f(x) = \sum_{Q \in \mathcal{S}} \langle f \rangle_{Q^{(m)}} \chi_Q(x), \] where $Q^{(m)}$ denotes the $m$-th dyadic parent of $Q$.

In Lerner's article, the main purpose of this formula was to prove a particular sharp weighted bound for Calderón-Zygmund operators previously known as the "$A_2$ conjecture" (and which became a Theorem when Tuomas Hytönen proved it.)

There was, however, a fundamental obstacle to obtaining such estimate. The "complexity" introduced by the fact that the average of $f$ is taken over the $m$-th parent of $Q$, instead of just $Q$ itself. This makes most naive estimates exponential in $m$, which is not enough to make the whole sum converge.

Lerner was able to circumvent this obstacle by taking norms (in an arbitrary Banach function space) and using duality; it turned out that the adjoint of the operator $\mathcal{A}_{\mathcal{S}}^m$ was easier to estimate (especially pointwise). After this duality trick one can reduce the complexity to $0$ and then the estimate follows by an easy argument.

There was a caveat, however. The method only works when duality is available, so the cases of $L^p$ spaces with $p<1$, or some Lorentz spaces, were out of reach.

The natural endpoints for many of these 'sharp weighted estimates' do not actually have a reasonable duality theory (or lack one altogether). Examples of these cases arise in the weak-type $(1,1)$ estimate for linear Calderón-Zygmund operators, the $k$-linear theory (where the natural endpoint is $L^{1/k}$), or in square-function estimates, to name a few. Hence, it was natural to ask whether an estimate in this spirit (estimating a singular integral in terms of positive dyadic shifts of complexity $0$) could work in the absence of duality. In fact some conditional results were obtained in Damián-Lerner-Pérez, Li-Moen-Sun and Lerner.

The purpose of this article is to give an answer to this question. We are able to give a pointwise estimate of the $m$-th complexity dyadic shifts by a factor (linear in $m$) times a $0$-complexity dyadic shift. The latter shift depends on the former, as well as on the function, but is uniform in all the relevant quantitative parameters.

The article gives the proof of the pointwise bound, as well as the construction of the bounding $0$-complexity shift. This lets us complete the conditional results cited above. Some open questions are left open.

@ARTICLE
{2014arXiv1409.4351C,
author = {{Conde-Alonso}, J.~M. and {Rey}, G.},
title = "{A pointwise estimate for positive dyadic shifts and some applications}",
journal = {ArXiv e-prints},
archivePrefix = "arXiv",
eprint = {1409.4351},
primaryClass = "math.CA",
keywords = {Mathematics - Classical Analysis and ODEs},
year = 2014,
month = sep,
adsurl = {http://adsabs.harvard.edu/abs/2014arXiv1409.4351C},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}